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International Conference on Complex Systems (ICCS2006)

On an irreducible theory of complex systems

Victor Korotkikh
Central Queensland University

Galina Korotkikh
Faculty of Business and Informatics, Central Queensland University

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     Last modified: August 13, 2006

Abstract
Complex systems profoundly change human activities of the day and may be of strategic interest. As a result, it becomes increasingly important to have confidence in the theory of complex systems. Ultimately, this calls for clear explanations why the foundations of the theory are valid in the first place. The ideal situation would be to have an irreducible theory of complex systems not requiring a deeper explanatory base in principle. But, the question arises: where such a theory could come from, when even the concept of spacetime is questioned as a fundamental entity?

In the talk we suggest that the concept of integers may take responsibility in the search for an irreducible theory of complex systems. In particular, we show that complex systems can be described in terms of self-organization processes of prime integer relations [1]. Starting with integers as the elementary building blocks and following a single organizing principle, such a self-organization process in control of a complex system makes up prime integer relations of one level from prime integer relations of the lower level. A prime integer relation is built as an inseparable object in the sense that if even one of the prime integer relations is not involved in its formation, then according to the organizing principle the rest of the prime integer relations can not make up an integer relation.

Remarkably, it is possible to geometrize the prime integer relations as two-dimensional patterns and isomorphically express the self-organization processes by the transformations of the patterns. As a result quantities of a prime integer relation and a complex system it describes can be defined by using quantities of a corresponding two-dimensional geometric pattern. For example, the area of the geometric pattern and the length of its boundary curve can specify two quantities of the complex system. Due to the isomorphism, the structure and the dynamics of a complex system are combined in our description. As self-organization processes of prime integer relations determine the correlation structure of a complex system, transformations of corresponding geometric patterns may characterize the dynamics of the complex system in a strong scale covariant form.

A global symmetry of a complex system can be observed as its geometric patterns reveal symmetries interconnected through the transformations. This global symmetry belongs to the complex system as a whole, but does not necessarily apply to its parts. The differences between the behaviors of the parts may be interpreted through the existence of gauge forces acting in their reference frames. As arithmetic fully determines the breaking of the global symmetry, there is no further need to explain why the resulting gauge forces exist the way they do and not even slightly different. We discuss how the gauge forces could be quantitatively classified.

Significantly, controlled by arithmetic only the self-organization processes of prime integer relations can describe complex systems by information not requiring further explanations. This offers the possibility of developing an irreducible theory of complex systems.

[1]. V. Korotkikh and G. Korotkikh, “Description of Complex Systems in terms of Self-Organization Processes of Prime Integer Relations”, arXiv:nlin.AO/0509008.




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